WASHINGTON—Tuesday’s split in a union-dues case highlights the practical difficulties facing an eight-member Supreme Court, as well as the stakes for the political parties as they battle over who will get to appoint a ninth justice.

Tie votes prevent the high court from fully executing its duty to set legal rules and precedent for the entire nation. While the court often decides cases on lopsided votes, about a fifth of its rulings in recent years have come on 5-4 tallies. Such cases now face the prospect of deadlock—as happened...